Particle system doesn’t align hair object to assigned vertex group
Your particle system is correctly setup, and your particles are being correctly emitted from your chosen vertex group.

There is only one eyelash particle being emitted though, I am guessing you will need more than one.

What your are seeing are Particle Children which are instanced from the parent one around a predefined radius. This radius is far bigger than the area you would want eyelashes to emit from.

Either reduce the radius, or disable children completely and increase the number of actual particles instead.

Turning off Rotation and emitting from Vertex instead might also help achieve a more tame look.